When periodic waves overlap, superposition can produce stable patterns of reinforcement and cancellation. This is interference.

Phase difference

For two sinusoidal waves of the same frequency, the result depends on their phase difference.

If they arrive in phase, crest aligns with crest and trough with trough. Their amplitudes reinforce: constructive interference.

If they differ in phase by $\pi$, crest aligns with trough. Equal amplitudes cancel completely: destructive interference.

Path difference

For waves of wavelength $\lambda$, a path difference

$$\Delta L=m\lambda$$

with integer $m$ produces constructive interference for sources initially in phase.

A path difference

$$\Delta L=\left(m+\frac12\right)\lambda$$

produces destructive interference.

Partial interference

Most phase differences lie between these extremes, so the resulting amplitude is neither the sum nor complete cancellation.

Coherence

A stable interference pattern requires a stable phase relationship. Sources whose relative phase changes unpredictably do not produce a persistent pattern when averaged over time.

Interference is not a new interaction between waves; it is the visible consequence of ordinary superposition combined with phase.